Analyzing Convergence and Rewriting Sequences: A Mathematical Approach

[Mappe]

I need the math tools to understand and analyze sequences and their convergence. I know for example that the fibonacci series can be rewritten such that we can calculate for example nr 153 without knowledge of previous numbers. What math subjects is needed to take care of more complicated sequences, rewriting them in an understandable fashion and understanding their convergence?

 

[S.G. Janssens]

Recursively defined sequences may behave in a surprisingly complicated fashion. Periodicity with arbitrarily large periods is possible, as well as a-periodic (but bounded) behavior. In all but the simplest cases, it is not possible to write down explicit formulas for their $n$th term. Perhaps you will enjoy An Introduction to Difference Equations by Elayd, Springer, 3rd edition, 2005.

 

[Samy_A]

You may want to have a look at generating functions.

 

[Ssnow]

Given a sequence $a_{n}$ in recursive way, so $a_{n}=f(a_{n-1})$ where $f$ is a function, you can start supposing that converge and solving the equation $l=f(l)$. The result (if there is a result) will be a candidate for your sequence $a_{n}$. For example given

$a_{n}=\frac{1+a_{n-1}}{a_{n-1}}$ with initial data $a_{0}=1$.

You can search a limit solving $l=\frac{1+l}{l}$ that gives you $l^2-l-1=0$ so $l=\frac{1}{2}\pm\frac{\sqrt{5}}{2}$. A good candidate is $\frac{1}{2}+\frac{\sqrt{5}}{2}$.